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```					                Lecture 9:
Exchange-correlation functional and
time-dependent DFT

Marie Curie Tutorial Series: Modeling Biomolecules
December 6-11, 2004

Mark Tuckerman
Dept. of Chemistry
and Courant Institute of Mathematical Science
100 Washington Square East
New York University, New York, NY 10003
Ψ0λ is the ground state wavefunction of a Hamiltonian
H=T+λW+V. Note that when λ=0. V=VKS and when
λ=1, V=Vext.
λ       λ

λ       λ
n(r1 )

1     g ( r  r ; n(r )) 
Exc [n]   dr n(r )   dr 0                  
 2          r  r        
Where fxc can be obtained by integrating over r’ with a
Change of variables s=r-r’.
Exc [n]  Exc [n]   ( J [ni ]  Exc [ni ])
SIC

i

ni (r )   i (r )
2
Extend the LDA form to include density gradients:

Exc [n]   dr n(r) f xc (n(r), n(r), 2 n(r))

Example: Becke, Phys. Rev. A (1988)

 2 (r )
Ex [n]  C x  dr n 4 / 3 (r )    dr n 4 / 3 (r )
1  6  sinh 1  (r )
n(r )
 (r )  4 / 3
n (r )

Functional form chosen to have the correct asymptotic behavior:

1                                              1
Ex   dr n(r)ex (r)                 lim ex (r)                 n(r)  e  r
2                                 r          r
Motivation for TDDFT

• Photoexcitation processes
• Atomic and nuclear scattering
• Dynamical response of inhomogeneous metallic systems.
The time-dependent Hamiltonian
Consider an electronic system with a Hamiltonian of the form:

H (t )  Te  Vee  V (t )
Where V(t) is a time-dependent one-body operator.

Our interest is in the solution of the time-dependent Schrödinger equation:


H (t ) (t )  i  (t )
t
 (t0 )   0
Let V be the set of time-dependent potentials associated and let
N be the set of densities associated with time-dependent solutions
of the Schrödinger equation. There exists a map G such that

G : V           N
The Hohenberg-Kohn Theorem
Since V(t) is a one-body operator:

(t ) V (t ) (t )   dr n(r, t )Vext (r, t )

Assume the potential can be expanded in a Taylor series:

1
Vext (r, t )        vk (r )(t  t0 ) k
k 0 k !

Suppose there are two potentials such that


Vext (r, t )  Vext (r, t )  c(t )
Then, there exists some minimum value of k such that

k
vk (r )  vk (r )  k Vext (r, t )  Vext (r, t ) t t  const
                            
t                                   0
The Hohenberg-Kohn Theorem

For time-dependent systems, we need to show that both the
density n(r,t) and the current density j(r,t) are different for the two
different potentials, where the continuity equation is satisfied:


n(r, t )  j(r, t )  0
t
n(r, t )    dr2  drNe  (r, s1 ,..., x Ne , t )
2

{ s}

j(r, t )    dr2  drNe * (r, s1 ,..., x Ne , t ) (r, s1 ,..., x Ne , t )   * (r, s1 ,..., x Ne , t )(r, s1 ,..., x Ne , t ) 
                                                                                                            
{ s}

For any operator O(t), we can show that:

d                             
i      (t ) O(t ) (t )  (t ) i O(t )  [O(t ), H (t )] (t )
dt                            t
The Hohenberg-Kohn Theorem
From equation of motion, we can show that


i  j(r, t )  j(r, t )t t  in(r, t0 ) Vext (r, t0 )  Vext (r, t0 ) 

t                          0

And, in general, for the minimal value of k alluded to above:

k 1
                                                           
k

i t            j(r, t )  j(r, t )t t    in(r, t0 ) i  Vext (r, t )  Vext (r, t ) t t  0

                                         0
 t                                  0

Hence, even if j and j’ are different initially, they will differ for times just later
than t0.
The Hohenberg-Kohn Theorem

For the density, since


 n(r, t )  n(r, t )   j(r, t )  j(r, t )  0
t
It follows that:
k 2
                                                           
k

 t            n(r, t )  n(r, t )t t    n(r, t0 )   Vext (r, t )  Vext (r, t ) t t  0
                      
                                        0
 t                                0

Therefore, even if n and n’ are initially the same, they will differ for times just
later than t0.

Hence, any observable can be written as a functional of n and a
function of t.

[n](t ) | O(t ) [n](t )  O[n](t )
Actions in quantum mechanics and DFT
Consider the action integral:

t             
A   dt ' (t ') i       H (t ') (t ')
t0             t '
Schrödinger equation results requiring that the action be stationary according to:

A
0
  (t )
Hence, if we view A as a functional of the density,

t                     
A[n]   dt ' [n](t ') i               H (t ') [n](t ')
t0                   t '
t
A[n]  B[n]   dt '  dr n(r, t )Vext (r, t )
t0

t                 
B[n]   dt ' [n](t ') i       T  Vee [n](t ')
t0                t '
Hohenberg-Kohn and KS schemes
Hohenberg-Kohn:

A              B
                 Vext (r, t )  0
 n(r, t )        n(r, t )

Kohn-Sham formulation: Introduce a non-interacting system with effective
potential VKS(r,t) that gives the same time-dependent density as the
interacting system. For a non-interacting system, introduce single-particle
orbitals ψi(r,t) such that the density is given by
Ne
n(r, t )    i (r, t )
2

i 1

KS action:
t                                                              1          n(r, t ')n(r ', t ') 
AKS [n]   dt '   (t ) i       Ts  (t ')   dr n(r, t ')Vext (r, t ')   dr dr '                        Axc [n]
t0
          t '                                             2               r r'           
Time-dependent Kohn-Sham equations
From  AKS /  n(r, t )  0 :

                 1                     
i   i (r, t )     2  VKS (r, t )  i (r, t )
t                2                     
n(r ', t )    Axc
VKS (r, t )  Vext (r, t )   dr '            
r  r '  n(r, t )

t
Axc [n]   dt '  dr n(r, t ) f xc (n(r, t ), n(r, t ))
t0
Linear response solution for the density

Strategy: Solve the Liouville equation for the density matrix to linear order.

H (t )  H 0  V (t )

(t ) V (t ) (t)   dr n(r, t)Vext ( r, t)

Quantum Liouville equation for the density operator ρ(t):


i  (t )  [ H (t ),  (t )]
t

Time-dependent density:

n(r, t )  (t ) (t) (t)
Linear response solution for the density

Write the density operator as:

 (t )  0   (t )
To linear order, we have


 (t )  i[ H 0 ,  (t )]  i[V (t ), 0 ]
t
Solution:

t
 (t )  i  dt ' e       iH0 (t t ')
[V (t '), 0 ]e
iH0 (t t ')
t0
Linear response solution for the density
To linear order:
t
 n(r, t )  0  (t ) 0   dr               dt  (r, t , r, t )Vext (r, t ')
t0

Where the Fourier transform of the response kernel is:

   (r)  m  m 0 (r)  0    0 0 (r)  m  m 0 (r)  0 
 (r, r ,  )    0 0                                                         
m        ( Em  E0 )  i              ( Em  E0 )  i     

Hence, poles of the response kernel are the electronic excitation energies.
from Appel, Gross and Burke, PRL 93, 043005 (2003).
Lecture Summary
• Adiabatic connection formula provides a rigorous theory of the
exchange-correlation functional and is the starting point of many
approximations.

• Generalization of density functional theory to time-dependent
systems is possible through generalization of the Hohenberg-Kohn
theorem.

• In linear response theory, the response kernel (or its poles) is the
object of interest as it yields the excitation energies.

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